Obtain a new door, you can fit it without having to take anything apart, it just snaps in.

Obtain a BG-E5 Battery Grip, these may be easier to find than a spare battery door. Second hand ones go for as little as $40. It clips to the bottom of the camera and allows you to use an extra battery, or even power the camera from a set of common AA batteries. It also doubles as a portrait grip. It does make your camera a bit larger though.

In case of emergency (I Need My Camera Right Now!) use an object such as a piece of toothpick or matchstick to keep the switch plunger depressed, and some tape to keep it in place. However, you should remove it before you take out the battery, and put it back only after you insert the battery again.

On mostly all cameras the battery compartment door maintains a constant pressure on the battery to ensure that it keeps in good electrical contact with the terminals in the camera. Without that pressure the camera would probably not start due to effectively a 'no power' situation or if it did start it might be intermittent in its operation.

On some cameras besides the above reason there are also connecting strips inside the cover which ensure that the batteries are connected in the correct manner (if inserted correctly) to provide the correct voltage to make the camera work. e.g. cameras that use 2 x 1.5V AA cells need to be joined in series connection to provide 3V. This is done by the cover.